Untitled - Aerobib - Universidad Politécnica de Madrid

13.6. CONTINUITY EQUATIONS 309
Chemical reactions take place only on the flame surface r = r l , which acts as
a sink for fuel and oxidizer and as a source for the combustion products. Therefore
one obtains for each different chemical species an equation similar to Eq. (13.10). In
this equation ρ must be substituted by the partial density ρ i = ρY i corresponding to
species A i , v by velocity v i = v + v di , where v di is the radial diffusion velocity of the
species, and m must be substituted by the partial flow m i of the species. Thus
4πr 2 ρY i v i = m i , (i = 1, 2, 3). (13.11)
Moreover, the following evident conditions must be satisfied
∑
Y i v i = v, (13.12)
i
∑
m i = m. (13.13)
i
Equation (13.11) is valid for each species A i , as long as r does not cross the flame.
Let us now apply these equations separately to the interior and exterior regions
of the flame.
Interior region r s ≤ r ≤ r l
In this region only fuel vapours and inert gases exist, since the incoming oxygen is
entirely consumed as it reaches the flame without crossing it.
Equation (13.11) applied to the fuel vapours A 1 gives
4πr 2 ρY 1 v 1 = m 1 = m, (13.14)
since on the droplet surface the mass flow m is the mass flow m 1 of the fuel produced
by evaporation.
By comparing (13.9) and (13.14)
Y 1 v 1 = v. (13.15)
Equation (13.12) gives
Y 1 v 1 + Y 2 v 2 = v. (13.16)
From Eqs. (13.15) and (13.16) results
v 2 = 0, (13.17)
that is to say within the interior region the inert gases are at rest and the fuel vapours
diffuse through them towards the flame.